For example:

The trigonometric function is given as, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}.\)

\(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}\)

Here, \(\displaystyle\alpha={225}^{\circ}.\)

Apply the half angle formula for the given trigonometric function, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) in which the angle \(\displaystyle{112.5}^{\circ}\) lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) which is negative in second quadrant.

Therefore in half angle formula , negative sign has to be put.

\(\displaystyle{\cos{{\left(\frac{\alpha}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{\alpha}}}}{{2}}}}\)

The value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}{i}{s},\)

\(\displaystyle{\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{{225}}}^{\circ}}}{{2}}}}\)

\(\displaystyle=\sqrt{{\frac{{{1}+{\left(-\frac{\sqrt{{2}}}{{2}}\right)}}}{{2}}}}\)

\(\displaystyle=-\sqrt{{{\left(\frac{{{2}-\sqrt{{2}}}}{{2}}\right)}}}\)

The trigonometric function is given as, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}.\)

\(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}\)

Here, \(\displaystyle\alpha={225}^{\circ}.\)

Apply the half angle formula for the given trigonometric function, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) in which the angle \(\displaystyle{112.5}^{\circ}\) lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) which is negative in second quadrant.

Therefore in half angle formula , negative sign has to be put.

\(\displaystyle{\cos{{\left(\frac{\alpha}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{\alpha}}}}{{2}}}}\)

The value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}{i}{s},\)

\(\displaystyle{\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{{225}}}^{\circ}}}{{2}}}}\)

\(\displaystyle=\sqrt{{\frac{{{1}+{\left(-\frac{\sqrt{{2}}}{{2}}\right)}}}{{2}}}}\)

\(\displaystyle=-\sqrt{{{\left(\frac{{{2}-\sqrt{{2}}}}{{2}}\right)}}}\)